Real-time scheduling is the theoretical basis of real-time systems engineering. Earliest Deadline First (EDF) is an optimal scheduling algorithm for uniprocessor real-time systems. Existing results on an exact schedulability test for EDF task systems with arbitrary relative deadlines need to calculate the processor demand of the task set at every absolute deadline to check if there is an overflow in a specified time interval. The resulting large number of calculations severely restricts the use of EDF in practice. In this paper, we propose new results on necessary and sufficient schedulability analysis for EDF scheduling; the new results reduce, exponentially, the calculation times, in all situations, for schedulable task sets, and in most situations, for unschedulable task sets. For example, a 16-task system that in the previous analysis had to check 858,331 points (deadlines) can, with the new analysis, be checked at just 12 points. There are no restrictions on the new results: each task can be periodic or sporadic, with relative deadline, which can be less than, equal to, or greater than its period, and task parameters can range over many orders of magnitude.